# ISEE Upper Level Quantitative : Geometry

## Example Questions

### Example Question #1 : Lines

Which is the greater quantity?

(a) The measure of an angle complementary to a  angle

(b) The measure of an angle supplementary to a  angle

(a) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(b) is greater

(b) is greater

Explanation:

Supplementary angles and complementary angles have measures totaling  and , respectively.

(a) The measure of an angle complementary to a  angle is

(b) The measure of an angle supplementary to a  angle is

This makes (b) greater.

### Example Question #3 : Lines

and  are complementary; .

Which is the greater quantity?

(A)

(B)

(B) is greater

It is impossible to determine which is greater from the information given

(A) is greater

(A) and (B) are equal

(B) is greater

Explanation:

Two angles are complementary if their degree measures total 90. Therefore,

Since , we can substitute, and we can solve for :

, making (B) the greater quantity.

### Example Question #4 : Lines

Note: diagram is not drawn to scale

Refer to the above diagram. If  , what is  ?

Explanation:

and  form a linear pair, so

Since , this can be rewritten as

, and the first equation can be rewritten as:

### Example Question #5 : Lines

Note: Figure NOT drawn to scale.

Which of the following pairs of numbers could give the measures of  and  ?

None of the pairs given in the other choices is correct.

Explanation:

The two angles form a linear pair and therefore their measures total . We check all of the pairs for this sum.

The correct pair is .

### Example Question #81 : Plane Geometry

Note: Figure NOT drawn to scale.

Refer to the above diagram. Which of the following triples could refer to the measures of , and  ?

All of the other responses are correct.

All of the other responses are correct.

Explanation:

The measure of an exterior angle of a triangle, which here is , is the sum of the measures of its remote interior angles, which here are   and . Therefore, we are looking for the sum of the first two angle measures to be equal to the third.

All four triples satisfy this condition.

### Example Question #7 : Lines

Note: figure NOT drawn to scale

The degree measure of  is five degrees greater than twice that of . Which is the greater quantity?

(A)

(B)

It is impossible to determine which is greater from the information given

(A) and (B) are equal

(A) is greater

(B) is greater

(A) is greater

Explanation:

The degree measure of  is five degrees greater than twice that of  - that is,

Since  and  form a linear pair,

, and by substitution,

This makes (A) greater.

### Example Question #8 : Lines

Note: Figure NOT drawn to scale

. What is  ?

Explanation:

and  form a linear pair and therfore, the the sum of their degree measures is

### Example Question #9 : Lines

Note: Figure NOT drawn to scale.

Evaluate .

Explanation:

, so

The measure of an exterior angle of a triangle, which here is , is the sum of the measures of its remote interior angles, which here are   and .

### Example Question #81 : Plane Geometry

What is the slope of a line that passes through points  and ?

Explanation:

The equation for solving for the slope of a line is

Thus, if  and , then:

### Example Question #81 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

What is the slope of a line that passes through points  and ?